Robin Small - Boscovich Contra Nietzsche

International Phenomenological Society

Boscovich Contra NietzscheAuthor(s): Robin SmallSource: Philosophy and Phenomenological Research, Vol. 46, No. 3 (Mar., 1986), pp. 419-435Published by: International Phenomenological SocietyStable URL: http://www.jstor.org/stable/2107313 .Accessed: 26/10/2013 11:32

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Philosophy and Phenomenological Research Vol. XLVI, No. 3, March i986

Boscovich Contra Nietzsche

ROBIN SMALL Monash University

An interest in physical theory is evident in many of the writings of Nietz- sche. So too is the influence on his ideas of an important thinker of the previous century whose significance in the development of modern science is perhaps still not fully recognized. Nietzsche encountered the work of Roger Joseph Boscovich at an early stage in his career, and was impressed by its relevance to his own concerns.' In it he found an alternative to traditional forms of atomism: for Boscovich, atoms were no longer solid par- ticles but instead 'points of matter', located in space but without any extension of their own. Boscovich argued that all the properties attributed to matter could be explained in terms of the properties of force. Each point of matter in his theory serves as the center of a field of force, which varies in a rather complicated way according to distance in order to give rise to the familiar phenomena of bodies and their behavior.

In later writings Nietzsche paid tribute to the theory of Boscovich as a discovery to be ranked with the contribution of Copernicus to human knowledge.' Just as Copernicus had shown the apparent immobility of the earth to be an illusion, so Boscovich revealed the apparent solidity of material substance to be a fiction based on the unreliable character of our senses. Materialistic atomism could thus be described as "one of the best refuted theories there are."3 In place of this doctrine we should instead see the world as consisting of 'centers of force'.4 It is the "states, alterations,

See K. Schlechta and A. Anders, Friedrich Nietzsche: Von den verborgenen Anfdngen seines Philosophierens (Stuttgart-Bad Cannstatt: Friedrich Frommann Verlag, i96z), pp. 127-53. KGW V/z, p. 54i and VII/z, p. z64. "KGW" here indicates Nietzsche, Kritische Gesamtausgabe: Werke, ed. G. Colli and M. Montinari (Berlin-New York: Walter de Gruyter, 1973- ).

3 Beyond Good and Evil, Sect. iz. Where available the English translations of Walter Kaufmann have been used. KGW VIII/3, pp. i63, i65 and i68 (The Will to Power, Sect. 567, 636, and io66, respectively). Nietzsche also uses the terms 'atoms of force': KGW VII/3, p. 284 (The Will to Power, Sect. 637, and 64z), and 'points of force': KGW VII/3, p. 378 and VIII/i, p. 90.

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combinations and developments"' of these centers of force, the variations in their "order and relation,"6 that according to Nietzsche constitute the processes of nature.

In a recent article, George J. Stack has analysed the influence of Bosco- vich on Nietzsche in some detail.7 I do not intend to repeat his account. My concern in this article is to do something quite different: I wish to discuss some ideas over which Boscovich and Nietzsche stand opposed. Of central importance here is the idea of the eternal recurrence of the same, a theme whose importance in Nietzsche's thought can hardly be overesti- mated. In many of his writings (though primarily those found in his unpublished notebooks) Nietzsche attempts to support this doctrine in terms of ideas based at least in part on his reading of Boscovich. Yet when we look into Boscovich's Theory of Natural Philosophy, we find that he describes a theory which is very similar to the one defended by Nietzsche, and offers what is clearly intended to be a refutation of it.8 It is this conflict that I intend to discuss. In addition I will mention a number of more recent discussions of the same theme, and show to what extent they continue the same dispute.

Boscovich's objections are raised against an argument which he presents in this way:

The combination of a finite number of terms are finite in number; but the combinations throughout the whole of infinite eternity must have been infinite in number, even if we assume that what is understood by the name of combinations is the whole series pertaining to so many thousands of years. Hence, in a fortuitous agitation of the atoms, if all cases happen equally, as is always the case in a long series of fortuitous things, one of them is bound to recur an infinite number of times in turn.9

The first reply he makes to this argument is an objection to any argument which relies on the concept of chance. Since everything has a cause, he claims, those who use such arguments "err in the fact that they consider that there is anything that is in itself truly fortuitous." Rather this is a term which we use when we are ignorant of the cause of something.

Having made this objection, Boscovich moves on to a different point which is rather more complex: "But, leaving that out of account, it is quite false to say that the number of combinations from a finite number of terms

5KGW V/z, P- 421- 6 Ibid., p. 398. 7 G. J. Stack, "Nietzsche and Boscovich's Natural Philosophy," Pacific Philosophical Quarterly 6z (i98i): 69-87.

8 Stack mentions Boscovich's rejection of the idea of eternal recurrence, but only as an "assumption." Ibid., p. 73.

9 Boscovich, A Theory of Natural Philosophy, trans. J. M. Child (Cambridge, Mass.: M.I.T. Press, i966), p. i9i.

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is finite, if all things that are necessary to the constitution of the universe are considered." Boscovich sees clearly that the crucial question concerns the way in which combinations of these elements are formed. If we imagine them as arranged along a single line and consider only their order to be significant, the argument will not be open to his objection. I readily acknowledge this much; that, if all the letters that go to form a poem of Virgil are shaken haphazard in a bag, and then taken out of it, and all the letters are set in order, one after the other, and this operation is carried on indefinitely, that combination which formed the poem of Virgil will return after a number of times, if this number is greater than some definite number.

However, the 'constitution of the universe' involves more than this. For one thing, it involves the arrangements of the points of matter within a space which has three dimensions, and moreover is infinitely divisible in each of them. Thus even if we do imagine these terms to be located along a single line, there will be an infinite number of locations for this line with respect to each of the three dimensions of space. Furthermore, when we take into account not only the order of the points of matter on this line but their distances from one another, we find that each of them can be arranged in an infinite number of possible ways with respect to the others. Considering the possible speeds and directions of motion of the points adds still more orders of infinity. Boscovich thus concludes: Hence, the number of combinations is infinite of an order that is immensely higher than the order of the infinity of instants of time; and thus, not only does it follow that not all the combinations are not bound to return an infinite number of times, but the ratio even of those that do not return is infinite, of a very high order.

We should understand that Boscovich's method of argument in terms of probability is not incompatible with his insistence that all such thinking is misguided. For this is an argumentum ad hominem. It is designed to show that if we do look at the world as a whole in terms only of the possible combinations of its elements, we must arrive at the conclusion that the occurrence of any particular combination is infinitely improbable. Bosco- vich wants to use this conclusion further in arguing for a divine provi- dence as the only cause which can, as he puts it, 'overcome' such an infinite improbability. We are not here concerned with this further line of thought, except to note how many of the assumptions of the argument for eternal recurrence are in fact accepted by Boscovich. He does for instance accept that the number of points of matter is finite.'0 This indeed provides him with another argument for the existence of God: how else can we account for the fact that there are this number of points rather than

Ibid., p. i92.

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some other?" He also accepts that the totality of matter occupies only a finite space. Yet he does not accept the first premise of the argument, that the combinations of a finite number of terms are finite in number. Nor does he accept the model which presents these combinations as deter- mined by chance, since this is incompatible with the principle that everything must have a definite cause.

Let us turn now to the argument used by Nietzsche as an attempted demonstration of his doctrine of eternal recurrence, and to some of the objections made against it in recent years. We shall see that these are to a large extent simply restatements of the objections made by Boscovich to the argument he describes. First of all we should remind ourselves of a typical version of the argument found in the notebooks of Nietzsche:

If the world may be thought of as a certain definite quantity of force and as a certain definite number of centers of force - and every other representation remains indefinite and therefore useless - it follows that, in the great dice game of existence, it must pass through a calculable number of combinations. In infinite time, every possible combination would at some time or other be realized; more: it would be realized an infinite number of times. And since between every combination and its next recurrence all other possible combinations would have to take place, and each of these combinations conditions the entire sequence of combinations in the same series, a circular movement of absolutely identical series is thus demon- strated.'1

Although we might have picked various other passages, this one is suitable for our purpose not just because it is very often cited, but because it brings out the presence in the argument not only of the element of chance, but also of the principle of determinism. Each makes an important, in fact essential, contribution to the argument. First the argument from probability is used to show that in an infinite time every possible state of force must recur. Then Nietzsche argues that if any state of force does recur, the states of force to which it is linked by this principle must also recur. As he puts it in Thus Spake Zarathustra, "Are not all things knotted together so firmly that this moment draws after it all that is to come?"'3 This in turn leads to a conclusion about eternal recurrence: "If only one moment of the world were to return - said the lightning - then everything would have to return. 14

" Ibid., p. '93. I KGW VIII/3, p. i68 (The Will to Power, Sect. io66). I' Thus Spake Zarathustra, "On the Vision and the Riddle." 14 KGW VII/I, p. 503. Cf. J. S. Mill, A System of Logic (London: Longman, 1970), p. 277:

"And if any particular state of the entire universe could ever recur a second time, all sub- sequent states would return too."

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Nietzsche is a determinist in this sense: he holds that the total state of the world at any one moment conditions the total state of the world at any other moment. The question is: how are we to understand this? Nietzsche parts company with traditional determinism in that he repudiates several common ideas about the significance of such determination. He denies that there are laws of nature which compel events to take one course rather than another.'5 He denies that causes bring about their effects by pushing and pressing other objects.'6 These are attempts to interpret nature in terms of everyday human activities which are assumed to be understood because of their familiarity. Nietzsche's answer is that they are not understood at all, and that the real nature of our own actions is as unknown to us as anything in the outside world. He concludes that a causal account of natural processes is not an 'explanation', if this expression is used to mean a reduction of the unfamiliar to the familiar, but only an 'interpretation'.

The mathematization of nature, however, is not touched by these criti- cisms. On the contrary, Nietzsche seems to be aware that this research program can proceed with more success if it is no longer hindered by mis- leading associations. To cease thinking of force as Druck und Stoss, and of atoms as solid Klfimpchen, is only to give up a model which has shown itself to be inadequate, not to invalidate the theory for which it was designed. It is after all the scientists themselves who have demanded the change: "The mathematical physicists could not use little clumps of atoms for their science: therefore they construct a world of points of force, with which one can calculate." 7 His model here is Boscovich, who he points out was not an idealist but a mathematician.'8 Nietzsche goes beyond Boscovich in some ways: his general attack on the notion of natural 'laws' would prevent him from talking of a 'law of force' in the way Boscovich does, and he criticizes the 'dynamic atom' as not yet far enough removed from the solid atom of traditional thinking.'9 But again all these changes do not alter his conviction that the world is calculable. The problem is only to find the model which best allows us to grasp the nature of this cal- culability.

One might be puzzled by Nietzsche's ability to combine a kind of determinism with an argument which compares becoming to a 'dice game', uses the idea of probability, and talks of "the enormously accidental char-

'5 Mixed Opinions and Maxims, Sect. 9; The Gay Science, Sect. io9. I6 KGW VII/3 p. 224. 17 KGW VII/3, p. 378. I8 Selected Letters of Friedrich Nietzsche, ed. and trans. C. Middleton (Chicago: Univer-

sity of Chicago Press, i969), p. i8z (Letter of zo March i88z). '9 KGW VIII/3, p. 50 (The Will to Power, Sect. 634).

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acter of combinations.""0 Ivan Soll makes this point when he writes:

Nietzsche's use of the idiom of probability theory, his saying that "in infinite time, every possible combination would at some time or other be realized," is not to be taken at face value. For, intertwined with his talk of "the great dice game of existence" and "the realization of every possible combination" we find the expression of a determinism that is at odds with the idea that changes in the world are to be understood on the model of a dice game and that makes it difficult to interpret Nietzsche as straightforwardly holding that every possible combination will in fact be realized."'

This argument has a strange feature. One can readily see that if it were valid, the refutation of determinism would be an easy task. We could accomplish such a refutation simply by pointing out the actual occurrence of dice games in the real world. Or conversely, any credibility granted to determinism would turn the actual occurrence of dice games into a philosophical problem. Neither of these alternatives seems very plausible, on the face of it. This suggests that the doctrine of determinism and the theory of probability need not be seen as 'at odds' with each other. After all, Laplace's classic statement of the principle of determinism is to be found in the second chapter of his Philosophical Essay on Probabilities, where it is followed by an account of probability in which dice games figure as an example more than once." Nietzsche's statement that "The hands of necessity shake the dice-box of chance"" is the metaphorical expression of a point which is not as paradoxical as it may at first seem.

Yet I think there is some force in these remarks of Soll. Perhaps it is this: determinism suggests the presence of patterns and regularities in natural processes, and the dice game image suggests the absence of any such patterns. Yet presence in reality does not imply presence within the scope of human knowledge. Nietzsche holds that the process of becoming is far more complex than anything within our powers of observation. For many phenomena we do admit that this is so - in the case of dice games, for example - but for others we suppose that we can identify cause and effect with some certainty. In Nietzsche's view this is simply an illusion. Even with the phenomena which are closest to us, our own mental states, our awareness extends only to a fragment of the whole process. We simplify

10 KGW VII/z, p.5 I. I. Soll, "Reflections on Recurrence: A Re-examination of Nietzsche's Doctrine, die Ewige Wiederkehr des Gleichen," in Nietzsche: A Collection of Critical Essays, ed. R. Solomon (New York: Doubleday Anchor, 1973), p. 328.

ii Laplace, A Philosophical Essay on Probabilities, trans. F. W. Truscott and F. L. Emory (New York: Dover, 1952), p. 3: "Present events are connected with previous ones by a tie based upon the evident principle that a thing cannot occur without a cause which pro- duces it."

23 The Dawn, Sect. 130.

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and thus misrepresent what happens in fitting it into our set of concepts, and in proceeding to use these as the model for an understanding of the processes of 'outside' reality. The real patterns in both cases remain unknown to us. From this one can see why Nietzsche uses the dice game image; he wants us to extend the admission we make with respect to dice games to cover the whole of reality, and recognize every other process as equally beyond our power to calculate and predict with real certainty. But once again, this is not a denial of the principle of determinism. What gives Soll's argument its appeal is, I think, our tendency to place confidence in our knowledge of reality. From this standpoint determinism has always been taken as an optimistic doctrine, providing a reassurance that such confidence is not misplaced, and that our searches for causes will not be in vain. If this is an illusion (as Nietzsche alleges) it is at least a beneficent one, perhaps even one of those errors which he holds to be indispensable for the conduct of life. Certainly a recognition of 'the great dice game of existence' is not likely to be advocated for any practical advantage. But this concern is not particularly relevant to the present discussion. Nietz- sche's use of the principle of determinism in his argument does not assume that we in fact possess knowledge of even a single causal link. It is sufficient for him to claim that the total state of the world at each moment determines the total state of the world at each other moment; and there is no inconsistency between this and his talk of "the great dice game of existence. "

Boscovich's first objection to the theory of eternal recurrence was based on the claim that considerations of chance were inconsistent with a com- mitment to determinism. His second objection was quite different, because it depended on approaching the argument about chance on its own terms. I turn now to a closer examination of this approach.

Similar arguments put forward by later writers have been directed specifically against Nietzsche's theory of eternal recurrence. The best- known of all such arguments is that of Georg Simmel, first published in I907. Simmel uses a simple model to show that a finite number of elements need not have a finite number of combinations. 4 He asks us to imagine three wheels of equal size, rotating on the same axis. One point is marked on the circumference of each wheel, the three points are arranged in one straight line, and the wheels are set in motion. The second wheel rotates twice as fast as the first and the third at i/X times the speed of the first. Simmel claims that, however long the wheels turn, "the state of the three points from which the movement began can never recur in all eter-

24 Simmel, Schopenhauer und Nietzsche, third edition (Miinchen-Leipzig: Verlag von Duncker und Humblot, I 03), p. i83.



nity." Thus the finitude of the number of elements does not ensure that their state at any one moment will eventually be repeated.

One can see that this argument is valid by following a reductio ad absurdum procedure. Let us suppose that at some time the three points do return to their original line-up. We can tell when the first wheel returns to its initial position, because whenever this happens, and only when it happens, the points on the first and second wheels are aligned. Having noted this fact we can forget the second wheel, whose function is to enable us to identify complete revolutions of the first wheel without reference to outside objects." Now suppose that the third wheel too returns to its original position. Clearly the first and third wheels will each have revolved an integral number of times. If we call these numbers a and b respectively, then given our information about the relative speeds of the wheels, it follows that b = akr. But nr is an irrational number, which means that there are no two integers a and b for which the formula ar = a/b holds true. Thus our hypothesis that the original position can again be reached must be abandoned, because it has led to a contradiction.

This argument has been described as a "simple mathematical refutation. "z6 Yet it depends on the assumption that an irrational number such as ar can be used to measure something in the real world, in this case the speed of a moving object. Now in order to think of speed in this way, we must make the same assumption either about space or about time. Given the position of ar in the denominator of the expression used here, it seems to be serving as a measure of time. This recalls the premise of Boscovich's claim that the number of possible arrangements of unextended objects even within a finite part of space must be infinite. That argument rested on assuming the infinite divisibility of space; this one rests on assuming the infinite divisibility of time. Granting the validity of the reductio ad absurdum argument just discussed, then, we are still able to choose which of several premises to abandon in order to avoid contradiction. Our choice need not be the same as Simmel's.

Several recent writers have attempted to offer a different reply to Sim- mel's argument. Ivan Soll for instance writes: "Simmel's refutation depends upon the possibility of patterns of recombination being regulated so that certain repetitions are ruled out, but, if Nietzsche did indeed hold that the recombination was random, Simmel's arguments would lose

25 Simmel adds that a thread is held in place across the wheels in order to mark the starting position; if this remains it seems to me that the second wheel is superfluous.

z6 K. Jasper, Nietzsche: An Introduction to the Understanding of his Philosophical Activity, trans. C. F. Wallraff and F. J. Schmitz (Chicago: Henry Regnery Company, i965), p. 355.

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some of its force."17 In a similar way, Joe Krueger supports one interpretation of eternal recurrence by arguing that "The advantage of the varia- tion thesis is that in its reliance upon the random character of recombination it provides a method of escape from Simmel's argument against recurrence since that argument is dependent upon a deterministic universe."i28 Bernd Magnus too criticizes Simmel for the way in which "he has ruled out randomized configurations." He concludes: "The persua- siveness of Simmel's conclusion, that the number of configurations or combination of configurations is infinite, depends entirely on the introduction of a formula which defines the outcome in advance. The formula suggests, rather curiously, that the order of configurations is not subject to probability requirements. That is a particularly odd oversight when one attempts to offer a cogent counterexample."29

All these replies to Simmel rely on the contrast between chance or ran- domness on the one hand and determinism on the other. But whereas in Boscovich's first objection this contrast is used against the notion of chance, these writers turn the argument around. They criticize Simmel for setting up his example in a way that eliminates the element of chance alto- gether. It is true that Simmel does this. He assumes not only that determinism holds true, but also that the formula governing the course of events is available and can be stated. Yet does an introduction of chance lessen the force of his argument? To bring in this element we need deny only Sim- mel's second assumption. Let us suppose, then, that these three wheels are allowed to move away from their common axis or to move at different speeds, and that these variations are allowed to occur at random. The outcome is simply that the number of possible situations is increased by several orders of infinity. In other words, Boscovich's second objection immediately becomes applicable. The recurrence of the original combination is no longer ruled out by a principle of determinism, but it is ruled out by a principle of probability. In the original example, the recurrence of the same state is impossible; in our revised version it is merely infinitely improbable. This seems not to be much of an improvement. Can we overcome the problem by keeping the presence of chance to some minimum? Suppose we do so by allowing the speed of just one of the three wheels to vary at random, meanwhile keeping the wheel on the same axis as the others. Since only a single variable is now involved, only a single order of

27 Soll, art. cit., p. 328. z8 J. Krueger, "Nietzschean Recurrence as a Cosmological Hypothesis," Journal of the

History of Philosophy i6 (1978): 44z. 29 B. Magnus, Nietzsche's Existential Imperative (Bloomington-London: Indiana Univer-

sity Press, 1978), p. 93.



infinity is introduced into the range of possible combinations. But this is the same kind of infinity that we attribute to the length of time through which the movement takes place. In that case, an estimation of the probability of the recurrence of any one state is impossible. We are left with an indeterminate result, which neither proves nor disproves the theory of eternal recurrence.

We might note before leaving Simmel's argument that another attempt has been made to offer a reply, along lines different from those just considered. Arnold Zuboff argues that Simmel fails to consider the 'phenomenal character' of his example. He writes: "How many different phenomenal possibilities are there in Simmel's line-up of wheels? I would say that there can be only finite phenomenally discernible line-ups."3" Therefore, he concludes, although Simmel is correct in saying that the original configuration will never recur, it is also correct to say that some configuration will occur which is phenomenally indistinguishable from that one. Furthermore, whereas actual repetition would be "overwhelmingly unlikely," this phenomenal repetition will in contrast be "overwhelmingly likely."

This I think is a sound argument. Certainly our senses are limited in their ability to detect differences between situations. That being the case, the wheels in Simmel's example would at some time reach a position which would appear to us to be the same as their original position. For some of us, of course, that might happen quite soon. The question however is: what does this prove? It seems to me that it misses the point of the doctrine of eternal recurrence. Nietzsche speaks of the recurrence of the same, not of the similar, and he insists on a very sharp distinction between the two: "The similar is not a degree of the same: rather something totally different from the same.""i Further, he denies that any of the things we take to be the same are really the same. This is one of the basic illusions of our experience: that in order to organize it into a pattern which will allow us to cope with our environment, we identify things as the same when they are merely similar. Nietzsche asserts however that reality is a process of absolute becoming: "The tree is something new in every moment: the form is asserted by us because we cannot perceive the finest absolute movement. "3Z

Now it is quite striking that Nietzsche's insistences on this point are to be found in much the same places as his sketches of the doctrine of eternal recurrence. That is because the two are complementary: Nietzsche places

30 A. Zuboff, "Nietzsche and Eternal Recurrence." in Solomon, op. cit., p. 354. 31 KGW V/z, P- 4?3 3 Ibid., Pa 452-

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great importance on the 'sameness' of eternal recurrence precisely because he denies the sameness of anything else. He even offers an argument designed to prove that this is impossible: two things the same as each other and existing at the same time would have to have the same causes, and that in turn would imply an endless regress. "Thus we would have to assume that something the same had been present back into all eternity, despite every alteration of the general state and creation of new properties - an impossible assumption!"33 I take it that Nietzsche's reason for rul- ing out this assumption as impossible lies in his view that everything in reality is related to everything else, or in other words, that there is only one world. If we accept this assumption, we must reject the notion of any series of causes which would stand apart from the general course of the world. Thus in turn we must reject the claim that within one state of the world there can exist two things the same as each other.

It would not be true to say that Zuboff's comment on Simmel's argument denies any of these points. Why then should it be open to criticism? One might criticize it for introducing the notion of 'phenomenal' reality into the argument at all. The point is not just that this is out of place in a thought experiment, but that it is out of place in considering a concept which can be approached only in terms of a thought experiment. I refer again to the concept of sameness, as it figures in Nietzsche's thinking. As many commentators have pointed out, the eternal recurrence of the same is not an event for which evidence of the empirical kind can be provided. Nor can familiar kinds of experience be cited in its support. If for instance I could remember the occurrence in a previous cycle of the experience I have at this moment, it would not be the same experience after all, because it would contain this added element. In just the same way, I cannot foresee the future recurrence of this same experience. Nor therefore can I (as some writers seem to think) have any of the kinds of feelings about this recurrence which would presuppose such a cognition. Another interesting consequence is that common-sense questions about the criterion of personal identity, for instance, simply make no sense in this context. Such questions arise only for concepts which can be applied to experience in the way that has specifically been ruled out by Nietzsche for the concept of sameness. This is not to say that it is useless or meaningless, only that its use and meaning are rather different in kind.

For these reasons it seems to me that Zuboff's comment does not do justice to Simmel's argument, because it modifies the strictness of the concept of sameness. In this respect Simmel's argument is noticeably faithful to the doctrine which it attacks, and all the more effective as a result. Nor

33 Ibid., p. 421. See also ibid., p. 4z8.



does it seem right to criticize Simmel's example for being 'arbitrary' or for not being 'plausible', as if a thought experiment were bound to observe such requirements. From the point of view of possible arguments about probability, Simmel's example is relatively favorable to Nietzsche's case, as we have seen. My conclusion is that none of these replies directs itself to the truly important aspect of Simmel's argument. Before looking into this, however, I wish to mention one more recent criticism of Nietzsche's doctrine which bears a strong affinity to Boscovich's second objection.

Arthur Danto has rejected Nietzsche's argument for reasons which have to do with the notion of energy. (For the purpose of this discussion, I will follow Nietzsche in using the word 'force' as equivalent to 'energy', incorrect though this is in terms of modern usage.) He writes:

Imagine some conservative energy system the total energy of which has some finite number, say 6, where some of the energy is kinetic. Suppose again that the kinetic energy increases, so the potential energy decreases, but at a rate such that the first approaches 6 while the latter approaches o. These limits could be approached indefinitely without being reached, and there could in principle be an infinite number of 'states' of kinetic energy, having a different magnitude at every instant, without the recurrence of any single magnitude.34

In this example, the total state of energy at any one time is simply this kinetic energy and the potential energy that remains, and the relation between the two is just the relation between their respective quantities. Danto is arguing that since an infinite number of such measures are possible even within the stated limit, the number of total states of energy is also infinite. In a footnote he states the underlying principle of the argument: "It clearly doesn't follow from the fact that the sum is finite that there is a finitude of parts." It is the application of this mathematical truth to the case of energy that provides Danto's argument with its point. But again, the question of that application is not a purely mathematical one. Rather it is a question about the character of energy. Is it possible for a certain kind of energy to approach the zero limit indefinitely without reaching it? Or is there rather a quantum of energy which represents the limit of any such process? Many scientists in this century have held that there is such a quantum, and if they are correct, Danto's argument must be invalid. Whether any of Nietzsche's various remarks on the subject of force can be seen as making the same claim seems to me unclear. Certainly he denies that force can ever be divided into equal parts; this is one of his arguments against the possibility of a state of equilibrium.35 Equal division is only

3 A. Danto, "Nietzsche," in A Critical History of Western Philosophy, ed. D. J. O'Con- nor (New York: The Free Press, i964), p. 400. See also A. Danto, Nietzsche as Philoso- pher (New York: The Macmillan Company, i965), p. zo6.

3 KGW V/z, p. 4z8.

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one case, and does not rule out other divisions. Yet one might argue that if force were infinitely divisible, a division of any force into equal parts would always be possible. If it is not, then force cannot be infinitely divisible. Again, Nietzsche does say that "the world may be thought of as a certain definite quantity of force and as a certain definite number of centers of force. "6 From this it does not follow that each center of force always contains the same quantity of force. In Danto's example the total quantity of energy is constant, and so is the number of its components, but each of these varies in the quantity of energy it contains. If however Nietzsche does hold that every particular force is constant in quantity, the number of possible total states of force will again be finite. In that case, the constancy of the total amount of force would no longer be a basic assumption, but rather a consequence of the constancy of both the number of forces and their individual quantities of force. Nietzsche would agree with one feature of Danto's example, however: his distinction between potential and kinetic energy, and his assumption that the same energy can at one time be potential and at another time kinetic energy. This does not affect the main argument, because such changes are not alterations in the quantity of energy. Still, it is interesting to note how far this example is in accord with Nietzsche's approach to physical theory. This is a point which I will return to later in the discussion.

Turning now to the other objections based on the notion of infinite divisibility, we can see a certain difference between them. Boscovich's original objection is stated in terms of space, while Simmel's example appears to rely on the idea that time is infinitely divisible. These might seem to be exactly analogous arguments. However it is noticeable that in both Nietzsche's proof of eternal recurrence and in the argument described by Boscovich the significance of space is rather different from that of time. Nietzsche says nothing about space in most of his presenta- tions of this argument, but most commentators assume that what he means by 'combinations' is to be explained in terms of space. In Bosco- vich's argument, where space is mentioned, it is in order to specify what is meant by a 'combination' of terms. Time on the other hand is stated to be infinite as a premise of each argument, and its contribution is a much more direct one. So I think we have to deal with each case in its own right.

Let us return therefore to Simmel's argument. Would Nietzsche agree with the premise underlying the application of ar to the measurement of time? Does he regard time as infinitely divisible? The answer to this is unclear.37 Yet it would be in keeping with Nietzsche's approach to offer a

36 KGW VIII/3, p. i68 (The Will to Power, Sect. io66). 37 In his early sketch for a theory of 'time atoms', Nietzsche does say: "Time is not a contin-

BOSCOVICH CONTRA NIETZSCHE 431


reply to Simmel's objection along these lines: "If time were infinitely divisible, then even within a finite span of time there would be an infinite number of moments at which combinations of forces could take place. But there are only a finite number of such combinations. Therefore must not every one of them occur within, say, the next minute? Including the one which is the same as the present state of affairs? But we know that this does not happen. Therefore we must reject the assumption that time is infinitely divisible."

This is just a version of Nietzsche's own basic argument, turned around in much the same way as his argument against the possibility of a final state."8 It still depends however on the premise that the combinations of a finite number of terms are finite in number. We might choose to abandon this premise in order to avoid the false conclusion of the argument rather than the other one. That possibility brings us back again to the question raised by Boscovich about combinations of terms. It seems that we cannot avoid confronting the problem of understanding just how such combinations are constituted. That this is the crucial question is seen in Bosco- vich's account. As we noted, he mentions a situation in which the number of combinations of a finite number of terms is indeed finite, namely the experiment in which the letters forming a poem of Virgil are arranged and rearranged at random but always in a single order. Here the combination is formed in a simple way: each term comes either before or after each other one in the sequence, and the sum of these relations is all that is needed to define a particular combination. But according to Boscovich, the arrangements of points of matter in space involve relations which are not limited in this way, and therefore the same conclusion cannot be reached.

We can ask various questions about the relations between centers of force which Nietzsche thinks make them into combinations. Are they, for example, reflexive or symmetrical or transitive relations? One thing seems to be clear: they cannot be relations which admit of degrees. This point is made in an article on eternal recurrence by M. C. Sterling, who argues that we need make only the assumption that combinations are formed by "an all-or-none sort of interaction" to arrive at the conclusion that the number of combinations of a finite number of centers of force will be finite.39 This however is not a complete argument; in fact it is only a beginning. That there are no degrees of interaction does not imply that there is not

uum at all, rather there are only totally distinct time points, not a line." KGW 111/4, p. i8i. But he does not return to this theme in later writings, as far as I know.

38 See e.g. KGW V/z, p. 432. 39 M. C. Sterling, "Recent Discussions of Eternal Recurrence: Some Critical Comments,"

Nietzsche Studien 6 (1977): z65.

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more than one kind of interaction. Sterling assumes without argument that this is the case. He assumes too that any relation between centers of force must be a symmetrical one. Hence his statement: "According to this, there would be no other states of these force-centers over and above their being either combined or not combined." Such assumptions enable his argument to proceed to its conclusion, but their unexplained status also casts its plausibility into some doubt. In the rest of this discussion, I will offer some suggestions which may provide assistance in approaching these unanswered questions.

Nietzsche does give us some clues as to the kinds of relations between centers of force which constitute a particular combination. In general he calls them 'power-relationships'.40 It appears that a power-relationship is asymmetrical. What happens in such an interaction is that "the stronger becomes master of the weaker, in so far as the latter cannot assert its degree of independence."4' In other words, the relation is one of the control or domination of one force by another. Several features of this concept can be stated readily enough: for instance, that it is a transitive relation, or that whereas a force can control more than one other force, one force cannot be controlled by more than one force. Such points also open up the possibility of indirect as well as direct relations between forces, which would help us in understanding how Nietzsche can hold that everything is somehow related to everything else.

One interesting question here is: what is the significance of space in power-relationships? Nietzsche states one important fact: that every force has a certain range. Just as we see things only within a certain horizon, so too all our senses are active only within particular limits.42 "In the same way, every force has its sphere; it acts only so far and so strongly and only on this and that, not on anything else, a sphere of limitation."43 Similarly Nietzsche writes, "Even in the domain of the inorganic an atom of force is concerned only with its neighborhood."44 Without this limitation, he claims, a force would be indeterminate in one respect, and that would be something of an absurdity. Nietzsche does not however say that there is any other role of space in the interaction of forces. That is: he does not suggest that the relation of control or power varies in degree as a continu- ous function of distance. This being the case, the argument concerning space used by Boscovich is one which would not apply to Nietzsche's

40 KGW V/z, P. 224 and 259, VII/3, P. 387 and VIII/i, p. 133 (The Will to Power, Sect. 631).

4I KGW VII/3, P. 283 (The Will to Power, Sect. 630). 42 The Dawn, Sect. 117. 43 KGW V/i, p. 642. 44 KGW VII/3, P. 284 (The Will to Power, Sect. 637).



account. The point is incidentally closely linked with Nietzsche's denial of

infinite space. He writes that "Only on the false assumption of an infinite space, in which force evaporates, so to speak, is the final state an unpro- ductive, dead one. "h45 I take it that what Nietzsche means by 'evaporation' is a situation in which every force passes out of the range of every other force. If forces were arranged and rearranged at random within an infinite space, the occurrence of this situation at some time or other would be very probable. But in that case there would be no more interaction between forces, and a 'final state' would result of the kind which Nietzsche takes to be impossible for reasons expressed in a different argument: "If an equilibrium of force had been reached at any time at all, it would still exist: thus it has never come about. The present state of things contradicts the assumption."46

I now turn again to Nietzsche's description of a power-relationship as one in which "the stronger becomes master of the weaker." The words stronger' and 'weaker' here do not refer to the quantities of force con- tained on each side, for Nietzsche emphasizes that a small force can 'become master' of a much larger force. To understand what he means we must note the distinction between two different kinds of force, or rather two different forms which any particular force can take. Nietzsche speaks of 'latent force' and 'living force', following the terminology of contem- porary writers such as Robert Mayer.47 When a force 'becomes master' of another force, it controls the tendency of this force to assume one or other of these forms. That is: it determines whether a latent force remains as it is or turns into a living force, and whether a living force remains a living force or becomes a latent force. In the modern terminology used in Danto's example, these alterations are the transformation of potential energy into kinetic energy and vice versa. More generally, then, we can say that the strength of a force lies in its power over the accumulation or the discharge (Ausl6sung) of other forces.48 Nietzsche also makes the dis-

KGW VII/i, p. i i. Sterling comments on this passage that "the assumption of infinite space would render what I call 'all-or-none interaction' much less plausible." Art. cit., p. z66. This seems an arbitrary statement. In the example provided by Boscovich, in which certain letters are arranged along a line, the finite or infinite length of the line makes no difference to the relations between the letters. Yet these are 'all-or-none' relations in the sense that they do not obtain to this or that degree. Why should extending this to three dimensions make a difference?

4 KGW V/z, p. 432. 4 R. B. Lindsay, Julius Robert Mayer: Prophet of Energy (Oxford: Pergamon Press,

'973). 48 KGW V/z, pp- 349-5I, 390-9i and 439-40.

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tinction here between a 'directing force' and a 'driving force'. He points out that people often mistake one for the other: "One has mistaken the helmsman for the steam."49 They suppose that 'more power' implies 'more force'. On the contrary, it is the economical use of force which is the criterion of power in such interactions.5" The directing force which expends as little as possible of its own force, like the helmsman who directs the course of a ship with the touch of a hand, has the most power.5' The idea that this kind of cause has to be equal to its effect is simply a fallacy, based on the assumption that every cause is a driving force.

This brief sketch allows us to see that Nietzsche's idea of 'combinations' of centers of forces involves relations which are not as simple as one might at first think. Several forms of force have been mentioned, such as the latent and 'living' forms; and one might add the other distinctions between heat, motion, electricity and so on, which would make the total picture more complicated. The transformations of forces are of various kinds, and so too therefore are their interactions and combinations. Yet as far as our present discussion is concerned, the main question is just whether there are only a finite number of ways for such combinations to be formed. Nothing we have said rules this out. The distinction between several kinds of control exercised by one force over another does not imply that there are degrees of this relation. We can con- clude that there is as yet no reason to regard Boscovich's second objection as a refutation of Nietzsche's theory of eternal recurrence. The main value of his arguments, and of those recent argument which have proceeded along similar paths, is that they force us to look more closely into the structure of this remarkable doctrine.

4 The Gay Science, Sect. 3 6o. 50 KGW VIII/z, p. zoi (The Will to Power, Sect. 639), and VIII/3, p. 53 (The Will to

Power, Sect. 689). For a different use of this 'principle of economy', see R. Small, "Nietzsche's God", Philosophy Today z6 (i98z): pp. 48-49.

5I It should be noted that nothing at all is said in this discussion about the will to power, that is, about the drive to gain more and more power. This is another topic, and one which requires a different approach from the kind followed here. I mention it here only to prevent confusion.



Article Contentsp. 419p. 420p. 421p. 422p. 423p. 424p. 425p. 426p. 427p. 428p. 429p. 430p. 431p. 432p. 433p. 434p. 435

Issue Table of ContentsPhilosophy and Phenomenological Research, Vol. 46, No. 3 (Mar., 1986), pp. 357-530Front MatterThe Indeterminability of Time in "Sein und Zeit" [pp. 357-379]Parts and Pieces [pp. 381-400]On Saving Heidegger from Rorty [pp. 401-417]Boscovich Contra Nietzsche [pp. 419-435]The Role of Surfaces in an Ecological Theory of Perception [pp. 437-453]On Divine Foreknowledge and Bringing About the Past [pp. 455-469]Sensation and Scientific Realism [pp. 471-482]Critical NoticesReview: untitled [pp. 483-507]Review: untitled [pp. 507-522]Review: untitled [pp. 523-528]

Recent Publications [pp. 529-530]Back Matter

Robin Small - Boscovich Contra Nietzsche

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